A normally distributed data set has a mean of 115 and a standard deviation of 5. What percent of data falls between 100 and 115?


To find the percentage of data that falls between 100 and 115 in a normally distributed data set with a mean of 115 and a standard deviation of 5, we can use the properties of the normal distribution.

### Step 1: Convert the raw scores to Z-scores
The Z-score formula is:

\[
Z = \frac{X - \mu}{\sigma}
\]

Where:
- \( X \) is the value in the data set.
- \( \mu \) is the mean.
- \( \sigma \) is the standard deviation.

For \( X = 100 \):

\[
Z_{100} = \frac{100 - 115}{5} = \frac{-15}{5} = -3
\]

For \( X = 115 \):

\[
Z_{115} = \frac{115 - 115}{5} = \frac{0}{5} = 0
\]

### Step 2: Find the corresponding percentiles
Using the standard normal distribution table (or a calculator):

- A Z-score of 0 corresponds to the 50th percentile (or 0.5000 probability).
- A Z-score of -3 corresponds to a percentile close to 0.13th percentile (or 0.0013 probability).

### Step 3: Calculate the percentage
The percentage of data that falls between 100 and 115 is the difference between the two percentiles:

\[
\text{Percentage} = (0.5000 - 0.0013) \times 100\% = 49.87\%
\]

So, **approximately 49.87%** of the data falls between 100 and 115.

Would you like more details or have any questions? 

Here are some related questions to explore:

1. What percentage of data falls between 115 and 120?
2. How much data falls within one standard deviation of the mean?
3. What is the probability of selecting a data point greater than 120?
4. How would the percentage change if the mean was different?
5. What is the Z-score for a value of 125?

**Tip:** Understanding Z-scores is crucial in statistics as they allow you to compare scores from different distributions.

Learn how to calculate the percentage of data falling between 100 and 115 in a normal distribution with mean 115 and standard deviation 5. This problem involves using Z-scores and understanding the properties of the normal distribution.

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